randomly select $k$ distinct cells in the upper triangle (excluding the diagonal), and then fix them as $1$. The elements in the lower triangle are set accordingly. Of course $k\le \frac{n^2-n}{2}$.

all other elements are independent uniform random variables over $[0,1]$.

The question is

Any known result for the distribution of the minimum/maximum row sum?

It is easier to just look at one row. Let $c$ be the number of chosen cells to be fixed as $1$, then its distribution is $\frac{{\left( {\begin{array}{*{20}{c}}
{n - 1} \\
c
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
{\frac{{{n^2} - n}}{2} - (n - 1)} \\
{k - c}
\end{array}} \right)}}{{\left( {\begin{array}{*{20}{c}}
{\frac{{{n^2} - n}}{2}} \\
k
\end{array}} \right)}}$, which equals the probability of $c$ chosen cells in the first row. The sum of the remaining $n-1-c$ cells are i.i.d. uniform, and their sum is Irwin–Hall distribution.

However, I have trouble to find the distribution for min/max row sum, as the row sums are not even independent.